MIT OpenCourseWare http ocw mit edu Electromechanical Dynamics For any use or distribution of this textbook please cite as follows Woodson Herbert H and James R Melcher Electromechanical Dynamics 3 vols Massachusetts Institute of Technology MIT OpenCourseWare http ocw mit edu accessed MM DD YYYY License Creative Commons Attribution NonCommercial Share Alike For more information about citing these materials or our Terms of Use visit http ocw mit edu terms Appendix E SUMMARY OF PARTS I AND II AND USEFUL THEOREMS IDENTITIES Ax B C AB x C A x B x C B A C C A V o y V B Vy V A B V A V B V x A B V x A V x B V ov V P v vo V VA A VyB V A V A x B B V x A A V x B v v V V V x A 0 V x V 0 V x V x A V V A V 2 A V x A x A A V A lV A A V A B A V B B V A A x V x B B x V x A V x OA Vo x A OV x A V x A x B A V B B V A B V A El A V B THEOREMS b V k Al Divergence theorem Stokes s theorem r sA nda J V AdV A A dl V C x A n da S Vx daI bi Table 1 2 Summary of Quasi Static Electromagnetic Equations Differential Equations Magnetic field system V x H J V J 0 VXE 1 1 1 aB Integral Equations H dl B nda 0 B 1 1 21 SJ n da 0 1 1 22 f E dl Tt 1 1 20 J n da B n da 1 1 23 where E E v X B Electric field system VX E 0 1 1 11 1 1 12 V D Pf V J 1 1 14 at VXH J 1 aD 1 1 15 1 1 24 EgE dl 0 D D nda fvpp dV 1 1 25 Jf nda 1 1 26 H dlT d vpdV J n da where J J pfv H H v XD D n da 1 1 27 Table 2 1 Summary of Terminal Variables and Terminal Relations Magnetic field system Electric field system V C Definition of Terminal Variables Charge Flux qk f B nda A pdV Voltage Current f n da ik Vk fE di Terminal Conditions dqk dAk Sdt iA 4 i i ik 1 dt iN geometry AN geometry qk qk v 1 vv geometry vk vk ql1 qN geometry Table 3 1 Energy Relations for an Electromechanical Coupling Network with N Electrical and M Mechanical Terminal Pairs Electric Field Systems Magnetic Field Systems Conservation of Energy N J dWm N 31 f ijdAj j 1 j 1 N M e a dxj dWe 31 v dq j 1 dWi I b j 1 M NV c dW2 I A di I fe dx3 1 i 1 f e d j qj dv I Ly dx3 j 1 d t 1 Forces of Electric Origin j 1 M e a ANl Ax f M axj at i x 1 t e hf g he aWe ql q x 1 je x e VN a f x x 1 l X1 h Relation of Energy to Coenergy N N Wm w i ij j W e We vq 3 1 3 1 Energy and Coenergy from Electrical Terminal Relations Wm j N 2 0 d 0 4 k We 1 W N i i Wm 1 0i 0 xI x ij 1 di mi W v ql f q 0 q 0 x1 x dq 0 xj m dq 1 0 x x dv n P t q vl I vj1 0 0 j 1 o 1 The mechanical variables f and x can be regarded as thejth force and displacement or thejth torque T and angular displacement Q0 Table 6 1 Differential Equations Transformations and Boundary Conditions for Quasi static Electromagnetic Systems with Moving Media Differential Equations V x H J Magnetic 1 1 1 6 1 35 n X Ha Hb K 6 2 14 a 1 1 2 B B 6 1 37 n B Bb 0 V 1 1 3 J 6 1 36 n J Jfb V 1 1 5 E E vr x B 6 1 38 n X Ea Eb B p0 H M 1 1 4 M M 6 1 39 V x E 0 1 1 11 E E 6 1 54 n X E a Eb 0 6 2 31 V D p 1 1 12 D D 6 1 55 n D a Db rr 6 2 33 1 1 14 P Pf r J Jf pfv 6 1 56 J 6 1 58 aD V X H Jf 1 1 15 H H v x D 6 1 57 n X Ha Hb K van X n x Da Db D eoE P P P 6 1 59 0 aB Vx E field systems H H Boundary Conditions V B O0 systems Electric Transformations a 1 1 13 6 2 7 v K7 0 6 2 9 Ba Bb 6 2 22 n Jf a Jb V Kf V pfa fb at 6 2 36 6 2 38
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